For \(\cal {PT}\) -symmetric systems, the dynamics of OTOCs signals the Yang-Lee edge singularity [1]} of phase transition and displays the quantized response to external driven potential [2]}. It is now widely accepted that the non-Hermiticity is a fundamental modification to conventional quantum mechanics [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, since many systems such as optics propagation in “gain-or-loss” medium [10]}, [11]}, the electronics transport in dissipative circuit [12]}, [13]}, [14]}, and cold atoms in tailored magneto-optical trap [15]}, [16]}, [17]}, [18]}, [19]}, are described by non-Hermitian theory. The extension of Floquet systems to non-Hermitian regime even unveils rich physics [20]}, [21]}, [22]}, [23]}, [24]}. For example, the scaling of the spontaneous \(\cal {PT}\) -symmetry breaking and its relation with classical chaos are uncovered in a non-Hermitian chaotic system [25]}. The ballistic energy diffusion [26]} and quantized acceleration of momentum current [27]} are reported in a \(\cal {PT}\) -symmetric kicked rotor (PTKR) model. The quantum kicked rotor (QKR) and its variants provide ideal platforms for investigating fundamental problems, such as the quantum transport in momentum-space lattice [28]}, [29]}, the quantum-to-classical transition of chaotic systems [30]}, [31]}, as well as quantum thermalization in many-body systems [32]}. The operator growth and chaotic information scrambling in different variations of QKR are still open issue and deserve urgent investigations.
